As shown in [1]} (see also [2]}), equation (REF ) can be derived in the framework of
the multiphase modulation (Whitham) theory [3]}, [4]} without making any assumptions about the gas' density or the collision rate. The derivation in [1]} makes an extensive use of the spectral properties of the so-called finite-gap potentials [6]}, the quasi-periodic generalisations of \(N\) -soliton solutions of the KdV equation exhibiting the band Lax spectrum, \(\lambda \in \mathcal {S}_N \equiv \cup _{i=1}^{N+1} \gamma _i , \quad \gamma _i \cap \gamma _j= \emptyset , \ \ i \ne j\) , where the spectral bands \(\gamma _j \subset \mathbb {R}, j=1, 2, \dots , N\)  are finite intervals and \(\gamma _{N+1}\)  – a semi-infinite interval. The \(N\) -soliton limit of an \(N\) -gap solution is achieved by collapsing all the finite bands \(\gamma _j\)  into double points corresponding to the solitonic spectral values \(\lambda _j\) .
